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Journal de physique

The critical behaviour of the three-dimensional dilute Ising model : universality and the Harris criterion

Jian‐Sheng Wang, Debashish Chowdhury

Published Jan 1, 1989• 43 citations
Publisher

Abstract

Monte Carlo simulation of quenched site-diluted three-dimensional Ising model has been carried out using the Swendsen-Wang algorithm. The susceptibility exponent and the correlation length exponent are estimated to be γ = 1.52 ± 0.07, v = 0.77 ± 0.04, respectively, for all p c < p < 1, where p is the concentration of the spins. In contrast to some earlier work, our results strongly support the concept of universality.

DOI
10.1051/jphys:0198900500190290500

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Journal of Experimental and Theoretical Physics•Dec 1, 2004•35 citations
Monte Carlo simulation of quenched site-diluted three-dimensional Ising model has been carried out using the Swendsen-Wang algorithm. The susceptibility exponent and the correlation length exponent are estimated to be γ = 1.52 ± 0.07, v = 0.77 ± 0.04, respectively, for all p c < p < 1, where p is the concentration of the spins. In contrast to some earlier work, our results strongly support the concept of universality.
We apply numerical simulations to study the criticality of the 3D Ising model with random site quenched dilution. The emphasis is given to the issues not being discussed in detail before. In particular, we attempt a comparison of different Monte Carlo techniques, discussing regions of their applicability and advantages/disadvantages depending on the aim of a particular simulation set. Moreover, besides evaluation of the critical indices we estimate the universal ratio Γ+/Γ−Γ^+/Γ^-Γ+/Γ− for the magnetic susceptibility critical amplitudes. Our estimate Γ+/Γ−=1.67±0.15Γ^+/Γ^- = 1.67 \pm 0.15Γ+/Γ−=1.67±0.15 is in a good agreement with the recent MC analysis of the random-bond Ising model giving further support that both random-site and random-bond dilutions lead to the same universality class.
Abstract. The critical properties of the randomly site-diluted two-dimensional Ising model were studied using the Wang-Landau algorithm. The concentration of nonmagnetic sites was q = 0.1; the remaining sites were occupied by magnetic particles. The study was carried out in the appro-priate restricted but dominant energy subspaces. The main effort was focused on the specific heat and magnetic susceptibility by using the density of states for the model for lattices with linear size L = 20-120. The finite size scaling behavior of the specific heat and susceptibility was studied. The ratio of the critical exponents (α/ν) is negative (consistent with the Harris criterion), while the ratio (γ/ν) appears to assume its pure Ising model value (weak universality). 1.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
We present a finite-size scaling analysis of high-statistics Monte Carlo simulations of the three-dimensional randomly site-diluted and bond-diluted Ising model. The critical behaviour of these systems is affected by slowly decaying scaling corrections which make the accurate determination of their universal asymptotic behaviour quite hard, requiring an effective control of the scaling corrections. For this purpose we exploit improved Hamiltonians, for which the leading scaling corrections are suppressed for any thermodynamic quantity, and improved observables, for which the leading scaling corrections are suppressed for any model belonging to the same universality class. The results of the finite-size scaling analysis provide strong numerical evidence that phase transitions in three-dimensional randomly site-diluted and bond-diluted Ising models belong to the same randomly dilute Ising universality class. We obtain accurate estimates of the critical exponents, ν = 0.683(2), η = 0.036(1), α = −0.049(6), γ = 1.341(4), β = 0.354(1), δ = 4.792(6), and of the leading and next-to-leading correction-to-scaling exponents, ω = 0.33(3) and ω 2 = 0.82(8).
Usually, the impact of structural disorder on the magnetic phase transition in the 3D Ising model is analyzed within the framework of quenched dilution by a nonmagnetic component, where some lattice sites are occupied by Ising spins, while others are nonmagnetic. This kind of quenched dilution, according to the Harris criterion, leads to a change in the critical exponents that govern the asymptotics in the vicinity of the phase transition point. However, the inherent reason for the emergence of a new, random Ising model universality class is not the presence of a nonmagnetic component, but the disorder in structure of spin arrangement. To demonstrate this fact, in this paper we set up extensive Monte Carlo simulations of a random mixture of two Ising-like magnets that differ in spin length s and concentration c. In doing so, we analyze the effect of structural disorder per se without appealing to the presence of a nonmagnetic component. We support our numerical simulations with renormalization group calculations. Our results demonstrate the emergence of the 3D randomly diluted Ising model universality class in a random mixture of two Ising magnets. While the asymptotic critical exponents coincide with those known for the site-diluted 3D Ising model, the effective critical behavior is triggered by parameters s and c. The impact of their interplay is a subject of detailed analysis.
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.